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Teaching mathematics 教數(shù)學 Time for a ceasefire 暫時?；?div> Technology and fresh ideas are replacing classroom drill—and helping pupils to learn 科技和新鮮的想法正代替課堂訓練,，幫助學生們?nèi)W習 Feb 1st 2014 | SHANGHAI AND TEL AVIV | From the print edition 譯者：kate2016 IF THE world’s education systems have a common focus, it is to turn out school-leavers who are proficient in mathematics. Governments are impressed by evidence from the World Bank and others that better maths results raises GDP and incomes. That, together with the soul-searching provoked by the cross-country PISA comparisons of 15-year-olds’ mathematical attainment produced by the OECD, a club of mostly rich countries, is prompting educators in many places to look afresh at what maths to teach, and how to teach it. 如果說全球的教育系統(tǒng)都有一個共同焦點的話，那就是培養(yǎng)出精通數(shù)學的畢業(yè)生,。世界銀行和其他的數(shù)據(jù)表明,，數(shù)學更好可以提高GDP和收入，這給政府留下了深刻印象,。國際學生評估項目（PISA）對多國15歲學生的數(shù)學素養(yǎng)進行了比較,，引人深思。這些因素促進了很多地方的教育者重新思考數(shù)學教什么,，怎么教數(shù)學,。國際學生評估項目（PISA）由經(jīng)濟合作與發(fā)展組織（OECD）制作發(fā)起，該組織的成員國大多是富裕國家,。 Those countries languishing in the league tables fret about how to catch up without turning students off the subject with boring drill. Top performers, most of them Asian (see chart), fear that their focus on technical proficiency does not translate into an enthusiasm for maths after leaving school. And everyone worries about how to prepare pupils for a jobs market that will reward creative thinking ever more highly. 在排名表上日漸衰弱的國家擔憂如何才能讓學生趕上其他國家,，而又不需要用枯燥的訓練讓學生對課程失去興趣。表現(xiàn)最優(yōu)者大多數(shù)是亞洲人（見圖表）,，他們擔心在離開學校后,，對技術熟練能力的關注并不能轉(zhuǎn)化為對數(shù)學的熱愛。而且人人都在擔心怎樣讓學生們?yōu)榫蜆I(yè)市場做準備,。就業(yè)市場獎勵創(chuàng)造性思維的程度將會越來越大,。 Maths education has been a battlefield before: the American “math wars” of the 1980s pitted traditionalists, who emphasised fluency in pen-and-paper calculations, against reformers led by the country’s biggest teaching lobby, who put real-world problem-solving, often with the help of calculators, at the centre of the curriculum. A backlash followed as parents and academics worried that the “new math” left pupils ill-prepared for university courses in mathematics and the sciences. But as many countries have since found, training pupils to ace exams is not the same as equipping them to use their hard-won knowledge in work and life. 數(shù)學教育曾是一片戰(zhàn)場：20世紀80年代美國的“數(shù)學之戰(zhàn)”使傳統(tǒng)主義者和改革者對立。改革派由全國最大的教學游說團體領導,。傳統(tǒng)主義者強調(diào)紙筆計算的熟練,。改革者主張課程的中心是常在計算器的幫助下,，來解決實際問題。強烈的抵制隨之而來,，家長和學者擔心這個“新數(shù)學”會讓學生在大學課程中的數(shù)學和其他科學準備不足,。但是正如很多國家已經(jīng)發(fā)現(xiàn)的那樣，訓練學生考試考得好,，并不等于訓練他們在工作和生活中,，運用來之不易的知識。 Today’s reformers think new technology renders this old argument redundant. They include Conrad Wolfram, who worked on Mathematica, a program which allows users to solve equations, visualise mathematical functions and much more. He argues that computers make rote procedures, such as long division, obsolete. “If it is high-level problem-solving and critical thinking we’re after, there’s not much in evidence in a lot of curriculums,” he says. 如今的改革者認為新技術使得這個舊爭論變得多余,。改革者包括康拉德·沃爾夫拉姆（Conrad Wolfram）,，他在Mathematica工作。Mathematica是一款數(shù)學軟件,，能幫用戶解方程,，使數(shù)學函數(shù)可視化以及其他功能。他認為電腦使機械過程過時了,， 比如長除法,。他說：“如果我們追求的是高水準的解決問題和批判性思維，然而在大多數(shù)課程中關于這方面的證據(jù)真的不多,?！?div> Estonia’s government has commissioned Mr Wolfram’s consultancy in Oxfordshire to modernise maths courses for secondary-school pupils. Starting this month, it will pilot lessons built around open-ended problems which have no single solution. One example: “What’s the best algorithm for picking a romantic date?” (Possible answer: go on more dates with a lower quality threshold to maximise the chance of success.) Another: “Am I drunk?”, which leads into quantitative analysis involving body masses, rates of alcohol absorption and other variables. 愛沙尼亞政府委任沃爾夫拉姆（Mr Wolfram）在牛津郡做顧問，為中學生改革數(shù)學課程,。本月開始,，它將會試開課程。這些課程設立開放式問題,，沒有唯一的答案,。舉一個例子：“選到一個浪漫的約會對象的最佳算法是什么？”（可能的答案：降低門檻,、增加約會次數(shù),，使成功幾率最大化。）另一個例子：“我喝醉了嗎,？”,，這就會引向定量分析包括身體質(zhì)量，酒精攝入率以及其他可變因素,。 Some PISA stars are also seeking a new approach. Singapore’s government commissioned David Hogan, an Australian maths educator, to assess its syllabus. It wants pupils to be able to explain what they know to their classmates, and to apply it in unfamiliar situations. 一些國際學生評估項目（PISA）之星也在尋求新的方法,。新加坡政府委任澳大利亞數(shù)學教育家大衛(wèi)·霍根（David Hogan）來評估他們的教學大綱。他們想要學生能夠向他們的同學解釋他們所知道的的東西,，并且能在不熟悉的情況中運用這些知識,。 Israel is also experimenting, aware of the imminent end of the windfall provided by the arrival of many mathematicians from the former Soviet Union. Some Israeli maths lessons from primary school onwards will soon be taught using inexpensive tablets, an approach inspired by Shimon Schocken. An academic at Herzliya University, from 2005 he has created a large online group of “self-learners” who build computers and programs from scratch. Pupils are encouraged to visualise their calculations, not just to get the right answers. (Your correspondent, raised on algebra worksheets, found an iPad exercise that involved grouping sheep in holding pens embarrassingly hard.) 以色列的數(shù)學也在進行改革實驗。他們意識到,，很多前蘇聯(lián)數(shù)學家?guī)淼念~外收益即將結束,。一些以色列數(shù)學課程,，從小學開始將很快使用便宜的平板電腦教學。這個方法受到了希蒙· 肖肯（Shimon Schocken）的啟發(fā),。他是荷茲利亞大學的一位學者,，2005年他創(chuàng)立了一個大型在線自學者組織，這些自學者從零開始學習組建計算機和編寫程序,。學生被鼓勵使他們的計算可視化,，而不僅僅就是得到正確答案。（本文作者當年靠做習題學代數(shù),，發(fā)現(xiàn)一個在臨時圈養(yǎng)圈分羊的iPad練習令人尷尬的難,。） Andreas Schleicher, the statistician in charge of the PISA studies, agrees that many of the algorithms—rote procedures for solving familiar problems—that children have traditionally spent years mastering are now easy to “digitise, automate and outsource”. But he cautions against concluding that they are therefore no longer worth teaching. A third of pupils tested in Shanghai, which tops the PISA maths tables—and where pupils drill in mental arithmetic, start algebra young and learn formulae by heart—can apply their knowledge to novel and difficult problems, compared with 2-3% in America and Europe. 安德烈斯·施萊歇（Andreas Schleicher）是一位統(tǒng)計學家，負責國際學生評估項目（PISA）的研究,。他認同過去很多算法（解決相似問題的機械過程）孩子傳統(tǒng)上要花數(shù)年掌握,，現(xiàn)在這些算法很容易實現(xiàn)“數(shù)字化、自動化和外包化”,。但是他反對那種認為這些算法不再值得教的結論,。在上海被測試的學生中，有1/3的人能夠把他們的知識運用到新穎,、困難的問題中,。而在美國和歐洲這一比例為2-3%。上海在國際學生評估項目（PISA）數(shù)學排名第一,。上海的學生訓練掌握心算,，年幼開始學代數(shù),、背誦學習數(shù)學方程,。 Youngsters, in China and elsewhere, need to have the grounding to assess whether a computer-generated answer is close to right before they start to rely on whizzy technology. Then they can happily abandon the drill in long division and quadratic equations. For the majority of maths moderates, the good news is that both sides in the fruitless trench warfare between progressives and traditionalists look ready for a ceasefire. 中國或者其他地方的青少年,，在他們開始依賴高新技術之前,，需要基礎訓練來評估一個計算機產(chǎn)生的答案是否接近正確,。然后他們才會高興地擺脫長除法和二次方程的練習,。對于大多的數(shù)學改良派來說有個好消息,，那就是數(shù)學改革者和傳統(tǒng)主義者之間毫無結果的壕溝戰(zhàn),，使得雙方考慮暫時?；稹?div> From the print edition: International
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